:: deftheorem defines to_power ASYMPT_0:def 20 :

for F, G being FUNCTION_DOMAIN of NAT , REAL holds F to_power G = { t where t is Element of Funcs (NAT,REAL) : ex f, g being Element of Funcs (NAT,REAL) ex N being Element of NAT st

( f in F & g in G & ( for n being Element of NAT st n >= N holds

t . n = (f . n) to_power (g . n) ) ) } ;

for F, G being FUNCTION_DOMAIN of NAT , REAL holds F to_power G = { t where t is Element of Funcs (NAT,REAL) : ex f, g being Element of Funcs (NAT,REAL) ex N being Element of NAT st

( f in F & g in G & ( for n being Element of NAT st n >= N holds

t . n = (f . n) to_power (g . n) ) ) } ;